Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

Lokalkonvexe Unterräume in Topologischen Vektorräumen und Das ε-Produkt

It is well-known that the algebraic tensor product E Y of a not necessarily locally convex topological vector space E and a locally convex space Y can be identified with a subspace of the so-called -product EY (a space of continuous linear mappings from Y into E). So, whenever EY is complete, even the completed tensor product is (isomorphic to) a subspace of EY. As this occurs in many important cases, it is interesting to remark that, for each continuous linear operator u from a locally convex space F into E, there exists a locally convex U with continuous embedding jUE and a continuous linear map ûFU such that u=j·û. As main applications of a combination of these ideas, we obtain a characterization of the functions in as continuous functions with values in locally convex spaces (this gives new aspects for the intergration theory of Gramsch [5]) and a result extending a theorem in [6] on holomorphic functions with values in non locally convex spaces to arbitrary complex manifolds.     

Homogenization of Some Elasticity Problems with Rapidly Oscillating Convex Constraints on Displacements

The problem of homogenization is considered for an elastic body occupying a perforated domain = obtained from a fixed domain and an -contraction of a 1-periodic domain .     

Nonlocal problems in the theory of the motion equations of Kelvin-Voight fluids

For the motion equations of Kelvin-Voight fluids one proves: 1) a global theorem for the existence and uniqueness of a solution (v;{u_e}) of the initial-boundary value problem on the semiaxis t R⁺ from the class W ¹ (R⁺); W ₂ ² () H()) with initial condition v_o(x) W ₂ ² () H() when the right-hand side f(x, t) L(R ⁺; L₂()); 2) a global theorem for the existence and uniqueness of a solution (v; {u_l}) on the entire axisR from the classW ¹ (R; W ₂ ² () H()) when the right-hand side f(x, t) L(R; L₂()); 3) a global theorem for the existence of at least one solution (v; {u_l}), periodic with respect to t with period , from the class W ¹ (R ⁺; W ₂ ² () H()) when the right-hand side f(x, t) L(R ⁺; L₂()) is periodic with respect to t with period , and a local uniqueness theorem for such a solution; 4) a theorem for the existence and uniqueness in the small of a solution (v; {u_l}), almost periodic with respect to t R, from V. V. Stepanov's class S ¹ (R; W ₂ ² ()H()) when the right-hand side f(x, t) S(R; L₂()) is almost periodic with respect to t; 5) the linearization principle (Lyapunov's first method) is justified in the theory of the exponential stability of the solutions of an initial-boundary value problem in the space H() and conditions are given for the exponential stability of a stationary and periodic solution, with respect to t R, of the system (1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 146–185, 1990.     

Estimates of the asphericity of sections of convex bodies

We define the function (n, k) to be the infimum of all such that any bounded centrally symmetric convex body inR ⁿ possesses an -asphericalk-dimensional central section. It is proved that (3, 2)=2–1 and (n, n-1)n-1-1. Several related functions are defined and their values on the pairs (n, n-1) are estimated.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 76–79.     

Stability of Bounded Solutions of Differential Equations with Small Parameter in a Banach Space

For a sectorial operator A with spectrum (A) that acts in a complex Banach space B, we prove that the condition (A) i R = Ø is sufficient for the differential equation where is a small positive parameter, to have a unique bounded solution x for an arbitrary bounded function f: R B that satisfies a certain Hölder condition. We also establish that bounded solutions of these equations converge uniformly on R as 0+ to the unique bounded solution of the differential equation x(t) = Ax(t) + f(t).     

Extensions and selections in subspaces ofC(K)

LetK be a compact Hausdorff space and letFK be a peak interpolation set for a function algebraAC(K). Let be a map fromK to the family of all convex subsets of such that the set {(z, x)zK, x(z)} is open inK×C and such thatg(z)(z) (zK) for somegA. We prove that everyfC(F) satisfyingf(s)(s) (sF) (f(s)closure (s) (sF)) admits an extensionfAA} satisfyingf(z)(z) (zK) (f(z))}closure (z) (zK), respectively). We prove a more general theorem of this kind and present various applications which generalize known dominated interpolation theorems for subspaces ofC(K).     

On Logarithmic Smoothing of the Maximum Function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation g of f itself, where >0 denotes the approximation parameter. The one-parametric family g converges – relative to a compact subset – uniformly to the function f as tends to zero. Under nondegeneracy assumptions we show that the stationary points of g and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x() of stationary points for g , where each x() converges to the corresponding stationary point of f as tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve x().     

Stochastic search in a convex region

Summary A stochastic search strategy is proposed for locating a possibility mobile target in a bounded, convex region of the plane. The strategy is asymptotically minimax as 0 with respect to the time required to get within of the target. The proof involves the study of first passages to time-dependent boundaries by a certain semi-Markov process.Supported by NSF grant DMS 82-01723     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

Interface Fluctuations and Couplings in the D=1 Ginzburg–Landau Equation with Noise

We consider a Ginzburg–Landau equation in the interval [–^–, ^–], >0, 1, with Neumann boundary conditions, perturbed by an additive white noise of strength We prove that if the initial datum is close to an "instanton" then, in the limit 0⁺, the solution stays close to some instanton for times that may grow as fast as any inverse power of , as long as the center of the instanton is far from the endpoints of the interval. We prove that the center of the instanton, suitably normalized, converges to a Brownian motion. Moreover, given any two initial data, each one close to an instanton, we construct a coupling of the corresponding processes so that in the limit 0⁺ the time of success of the coupling (suitably normalized) converges in law to the first encounter of two Brownian paths starting from the centers of the instantons that approximate the initial data.     

Nonlocal problems in the theory of equations of motion of Kelvin-Voight fluids. 2

This article studies nonlocal problems for equations of motion of Kelvin-Voight fluids (2): 1) global solvability of initial-boundary-value problem (2)-(3) on halfaxisR ⁺ with free termf(x, t) S₂(⁺; L₂(0)) (see (4)); 2global solvability of system (2) on the entire axis R in the class of functions that are bounded as t±with free term f(x,t)S₂(; L₂());3) the existence of periodic solutions for system (2) that are periodic in t with period with free term,f(x,t)L₂((0,); L₂());4) the existence of solutions of system (2) that are almost periodic in t with free term f(x,t)S₂(, L₂()).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 111–124, 1990.     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

Efficiency and the uniform linear minorization of convex functions

LetX be a Banach space, and let {f _i:iI} be a family of proper lower semicontinuous convex functions defined onX, each of whose epigraphs meets a fixed bound subset ofX×. We say that {f _i:iI} is uniformly linearly minorized if there exists a positive scalar such that for alliI andxX, we havef _i(x)–(1+x). We present two very different characterizations of uniform linear minorization for such a family. Using one of these, we show that either strong or weak epi-convergence of a sequence of convex functions at some point in the effective domain of the limit implies, uniform linear minorization for the entire sequence.With 1 Figure     

Distance functional and suprema of hyperspace topologies

Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the weakest topology on CL(X) such that A - d(x, A) is continuous for each x X and each admissible metric d. We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) - {d(ta, b): a A, b B} is continuous on CL(X) × CL(X).Visiting the University of Minnesota.Visiting California State University, Los Angeles.     

Stability of Regularized Bilevel Programming Problems

A bilevel programming problem S is considered. First, sufficient conditions of minimal character are given on the data of the problem in order to guarantee the lower semicontinuity of the marginal function of the upper level problem. Then, for >0, a regularized problem S() is considered for which continuity of the regularized marginal function and convergence of the approximate value, as goes to zero, are obtained. Moreover, under perturbations on the data, convergence results for the perturbed marginal functions and the solutions to the problem S _n() are given for any >0.     

On the complex zeros of solutions of linear differential equations

Summary A classical result (see R.Nevanlinna, Acta Math.,58 (1932), p. 345) states that for a second-order linear differential equation, w + P(z) w + Q(z) w=0, where P(z) and Q(z) are polynomials, there exist finitely many rays, arg z=_j, for j=1,..., m, such that for any solution w=f(z) 0 and any > 0, all but finitely many zeros off lie in the union of the sectors ¦ arg z - _j¦ < for j=1,..., m. In this paper, we give a complete answer to the question of determining when the same result holds for equations of arbitrary order having polynomial coefficients. We prove that for any such equation, one of the following two properties must hold: (a) for any ray, arg z=, and any > 0, there is a solution f 0 of the equation having infinitely many zeros in the sector ¦arg z - ¦ <, or (b) there exist finitely many rays, arg z=_j, for j= 1,..., m, such that for any >0, all but finitely many zeros of any solution f 0 must lie in the union of the sectors ¦ arg z - _j¦ < for j=1, ..., m. In addition, our method of proof provides an effective procedure for determining which of the two possibilities holds for a given equation, and in the case when (b) holds, our method will produce the rays, arg z=_j. We emphasize that our result applies to all equations having polynomial coefficients, without exception. In addition, we mention that if the coefficients are only assumed to be rational functions, our results will still give precise information on the possible location of the bulk of the zeros of any solution.This research was supported in part by the National Science Foundation (DMS-84-20561 and DMS-87-21813).     

Asymptotic Behavior of Higher-Order Perturbations: Scaling Functions of the O(n)-Symmetric φ4-Theory in the (4−ε)-Expansion

We use an instantonic approach to calculate the asymptotic behavior of higher orders of the (4–)-expansion for the scaling function of the pair correlator of the O(n)-symmetric ⁴-theory in the minimal subtraction scheme. Our results differ substantially from the known exact expression for the ³ order of the expansion of the scaling function in the small- domain.     

Inverse Problems in the Theory of Singular Perturbations

First, in joint work with S. Bodine of the University of Puget Sound, Tacoma, Washington, USA, we consider the second-order differential equation ² y'=(1+² (x, ))y with a small parameter , where is analytic and even with respect to . It is well known that it has two formal solutions of the form y^±(x,)=e^±x/h^±(x,), where h^±(x,) is a formal series in powers of whose coefficients are functions of x. It has been shown that one (resp. both) of these solutions are 1-summable in certain directions if satisfies certain conditions, in particular concerning its x-domain. We show that these conditions are essentially necessary for 1-summability of one (resp. both) of the above formal solutions. In the proof, we solve a certain inverse problem: constructing a differential equation corresponding to a certain Stokes phenomenon. The second part of the paper presents joint work with Augustin Fruchard of the University of La Rochelle, France, concerning inverse problems for the general (analytic) linear equations ^r y' = A(x,) y in the neighborhood of a nonturning point and for second-order (analytic) equations y' - 2xy'-g(x,) y=0 exhibiting resonance in the sense of Ackerberg-O'Malley, i.e., satisfying the Matkowsky condition: there exists a nontrivial formal solution such that the coefficients have no poles at x=0.